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Theorem acneq 8425
 Description: Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acneq AC AC

Proof of Theorem acneq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2539 . . . 4
2 oveq2 6293 . . . . 5
3 raleq 3058 . . . . . 6
43exbidv 1690 . . . . 5
52, 4raleqbidv 3072 . . . 4
61, 5anbi12d 710 . . 3
76abbidv 2603 . 2
8 df-acn 8324 . 2 AC
9 df-acn 8324 . 2 AC
107, 8, 93eqtr4g 2533 1 AC AC
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379  wex 1596   wcel 1767  cab 2452  wral 2814  cvv 3113   cdif 3473  c0 3785  cpw 4010  csn 4027  cfv 5588  (class class class)co 6285   cmap 7421  AC wacn 8320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-acn 8324 This theorem is referenced by:  acndom  8433  dfacacn  8522  dfac13  8523
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